Introduction to Industrial Automation

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Appendix A: Arithmetic 

Systems

A.1   Introduction to Arithmetic Systems

Although knowledge related to arithmetic systems is the subject of books with mathematical or 

computer content, some basic issues of arithmetic systems are included in this book because it is 

absolutely necessary for an understanding of the operation and programming of PLCs. On the 

other hand, the reader’s knowledge of arithmetic systems is usually satisfactory from basic com-

puter courses, and it is much more necessary in using PLCs than personal computers. Even the 

most numerate user of a PC does not often come into contact with the concept of a “bit” since 

there is no such need. In contrast, in the world of PLCs, the programmer of an automation system 

starts with the utilization of bits, continued by integers, real numbers, or hexadecimal numbers, 

and ends again with the utilization of bits. The basic reason for this is the fact that the majority of 

devices that a PLC controls are of a binary status (binary digits or bits, such as 0 or 1, ON or OFF). 

Moreover, there are indication devices operating in a BCD code, where they are used, among oth-

ers, in the octal system for the input/output address and corresponding program instructions for 

bit to bit operations, etc. Therefore, someone who knows the arithmetic systems and codes better 

will have an easier time programming and making use of even the most specialized instructions.

An arithmetic system is a code of symbols which are assigned a quantity. From the moment the 

code has been defined and memorized, it can be used to measure any quantity. The decimal arith-

metic system uses ten basic (primordial) symbols or digits, where each digit represents a certain 

quantity. When several digits are grouped together, then larger quantities can be measured. The 

base of an arithmetic system is called the number of the primordial symbols or digits of the system. 

All arithmetic systems use position weighting to express the significance of each digit in a group.